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System of Particles and Rotational Motion

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Physics Part I

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CBSE Gujarat Board Haryana Board

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Class 10 Class 12
A non-uniform bar of weight W is suspended at rest by two strings of negligible weight as shown in Fig.7.39. The angles made by the strings with the vertical are 36.9° and 53.1° respectively. The bar is 2 m long. Calculate the distance d of the centre of gravity of the bar from its left end.

The free body diagram of the bar is shown in the following figure.

Length of the bar, l = 2 m 

Tand T2 are the tensions produced in the left and right strings respectively.

At translational equilibrium, we have

T1 Sin 36.90 = T2 Sin 53.1

      straight T subscript 1 over straight T subscript 2 = 4 over 3 

⇒      T1 = open parentheses 4 over 3 close parentheses T2
For rotational equilibrium, on taking the torque about the centre of gravity, we have

      T1 (Cos 36.9) × d = T2 Cos 53.1 (2 - d) 

            T1 × 0.800 d = T2 × 0.600 (2 - d

  open parentheses 4 over 3 close parentheses × T2 × 0.800d = T2 (0.600 × 2 - 0.600d

          1.067d + 0.6d = 1.2 

∴                         d = 1.2 / 1.67 

                             = 0.72 m 

Hence, the centre of gravity of the given bar lies 0.72 m from its left end. 

Define centre of mass.

Centre of mass of a body or a system of bodies is a point at which the entire mass of the body or system is supposed to be concentrated. 

Is it necessary for centre of mass to lie within the body?

No, centre of mass needs not to lie within the body. It is not necessary that the total mass of the system be actually present at the centre.

The position of the centre of mass is calculated using the usual Newtonian type of equations of motion. 

What is the need of centre of mass?

Newton’s second law of motion is strictly applicable to point masses only. To apply the Newton's law of motion to rigid bodies, the concept of centre of mass is introduced.

The concept of centre of mass of a system enables us to discuss overall motion of the system by replacing the system by an equivalent single point object. 

Is it necessary that there should be matter at the centre of mass of system?

No, it is not necessary that there be matter at the centre of mass of the system.

For e.g., if two equal point masses are separated by certain distance, the centre of mass lies at the mid point of two point masses and there is no mass at that point.

What is the significance of defining the center of mass of a system?

The motion of n particle system can be reduced to one particle motion.

An equivalent single point object would enable us to discuss the overall motion of the system.